📚 Year 12 WJEC Further Mathematics: Core Knowledge Review | Year 12 WJEC 进阶数学：核心知识点梳理
Year 12 WJEC Further Mathematics builds on the pure and applied mathematics studied in Year 12 Mathematics. The central compulsory unit is Further Pure Mathematics 1 (FP1), which deepens core techniques and introduces new areas such as complex numbers, matrices, vectors, hyperbolic functions, polar coordinates and proof by induction. Alongside FP1, students choose two applied modules from Further Mechanics, Further Statistics, or Decision Mathematics. This revision guide focuses on the essential concepts and methods from FP1, providing a clear bilingual summary of the skills you need to master.
Year 12 WJEC 进阶数学建立在 Year 12 数学课程的基础之上。核心必修单元是 Further Pure Mathematics 1 (FP1)，它深化了核心技巧并引入复数、矩阵、向量、双曲函数、极坐标和归纳证明等新内容。除 FP1 外，学生还需选修两个应用模块，可选方向包括进阶力学、进阶统计学或决策数学。本篇复习指南聚焦 FP1 的基本概念和方法，以清晰的双语形式梳理你需要掌握的核心技能。
1. Proof and Mathematical Induction | 证明与数学归纳法
Proof techniques in FP1 include direct proof, proof by contradiction, proof by exhaustion, and mathematical induction. Direct proof proceeds from given assumptions to the conclusion through logical steps. Proof by contradiction assumes the negation of the statement and works until a contradiction arises. Proof by exhaustion simply checks every possible case when the number of cases is finite.
FP1 中的证明技巧包括直接证明、反证法、穷举法和数学归纳法。直接证明从已知假设出发，通过逻辑步骤得出结论。反证法假设命题的否定成立，推导直至出现矛盾。穷举法在情况数量有限时逐一检验所有可能的情况。
rue for n = k, and then prove the statement for n = k + 1 using the assumption. Common applications include proving summation formulas, divisibility relationships, and matrix power identities.
数学归纳法用于证明涉及自然数的命题。步骤分为三步：证明基础情况（通常 n = 1）；假设 n = k 时命题成立；利用该假设证明 n = k + 1 时命题也成立。常见应用包括证明求和公式、整除关系和矩阵幂的恒等式。
Example induction: Show Σ r³ = ¼ n² (n + 1)² for all n ∈ ℕ
证明示例：对所有自然数 n，证明 Σ r³ = ¼ n² (n + 1)²
2. Complex Numbers | 复数
ate is z* = a − b i. Operations of addition, subtraction and multiplication use the ordinary rules of algebra, replacing i² with −1. Division is carried out by multiplying numerator and denominator by the conjugate of the denominator.
复数表示为 z = a + b i，其中 a, b 为实数且 i² = −1。实部 Re(z) = a，虚部 Im(z) = b。复共轭为 z* = a − b i。加减与乘法运算遵循代数规则，并将 i² 替换为 −1。除法通过分子分母同乘分母的共轭来实现。
The modulus of z is |z| = √a² + b² n θ = b/a (adjusted for the quadrant). The polar form is z = r (cos θ + i sin θ), with r = |z|. Euler's identity gives the exponential form: z = r eiθ. De Moivre' n be used to find powers of complex numbers and to derive trigonometric identities. To find the n-th roots of a complex number, use the formula z1/n = r1/n [cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)] for k = 0, 1, ..., n−1.
z 的模为 |z| = √a² + b²，辐角 arg(z) = θ 满足 tan θ = b/a（需根据象限调整）。极坐标形式为 z = r (cos θ + i sin θ)，其中 r = |z|。欧拉恒等式给出指数形式：z = r eiθ。棣莫弗定理指出，对整数 n 有 (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。该定理可用于求复数的乘方和导出三角恒等式。要求复数的 n 次方根，使用公式 z1/n = r1/n [cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)]，k = 0, 1, ..., n−1。
3. Matrices and Linear Transformations | 矩阵与线性变换
e but not commutative. The determinant of a 2×2 matrix M = [[a, b], [c, d]] is det(M) = ad − bc. For a 3×3 matrix, the determinant is found by expansion by minors. A matrix is invertible only if its determinant is non-zero. The inverse of a 2×2 matrix is (1/det) [[d, −b], [−c, a]].
你必须能够对合适阶数的矩阵进行加、减、乘法运算，包括 2×2 和 3×3 矩阵。矩阵乘法满足结合律但不满足交换律。2×2 矩阵 M = [[a, b], [c, d]] 的行列式为 det(M) = ad − bc。对于 3×3 矩阵，行列式可通过代数余子式展开求得。只有行列式不为零的矩阵才可逆。2×2 矩阵的逆矩阵为 (1/det) [[d, −b], [−c, a]]。
ion in the x-axis: [[1, 0], [0, −1]]; and scaling. A system of simultaneous linear equations can be written as Ax = b and solved using the inverse matrix x = A⁻¹b, provided A is invertible.
矩阵可表示二维和三维空间中的线性变换。常见二维变换有：旋转 θ 角矩阵 [[cos θ, −sin θ], [sin θ, cos θ]]；关于 x 轴的反射矩阵 [[1, 0], [0, −1]]；以及缩放变换。线性方程组可以写成 Ax = b 的形式，并在 A 可逆时通过逆矩阵求解 x = A⁻¹b。
4. Partial Fractions and Inequalities | 部分分式与不等式
To decompose a rational expression into partial fractions, first factorise the denominator. For distinct linear factors (ax + b)(cx + d), write the expression as A/(ax + b) + B/(cx + d) and solve for A and B. For repeated factors (ax + b)², use A/(ax + b) + B/(ax + b)². If there is an irreducible quadratic factor, include (Ax + B)/(ax² + bx + c).
要将有理式分解为部分分式，首先对分母进行因式分解。对于不同线性因式 (ax + b)(cx + d)，将表达式设为 A/(ax + b) + B/(cx + d) 并解出 A 和 B。对于重复因式 (ax + b)²，应使用 A/(ax + b) + B/(ax + b)²。如果存在不可约二次因式，则需加入 (Ax + B)/(ax² + bx + c)。
al values where the expression equals zero and test intervals between them. For rational inequalities like f(x)/g(x) ≥ 0, multiply both sides by the square of the denominator (which is positive unless zero) or use a sign table. Absolute value inequalities are tackled by considering the piecewise definition or squaring both sides.
解多项式和有理不等式需要细致的符号分析。对于如 (x − 2)(x + 1) > 0 的多项式不等式，找出表达式为零的临界值并检验区间。对于有理不等式 f(x)/g(x) ≥ 0，两边乘以分母的平方（除非分母为零，否则该平方为正）或使用符号表。绝对值不等式可通过考虑分段定义或两边平方来处理。
5. Series and Summations | 数列与求和
n+1)/2]². These can be combined with algebraic rules to sum more complicated polynomial series. Remember that Σ (constant) = n × constant.
FP1 要求你使用并拓展标准求和结果：Σ r = n(n+1)/2，Σ r² = n(n+1)(2n+1)/6，Σ r³ = [n(n+1)/2]²。结合代数运算法则，可以得出更复杂的多项式级数的和。注意 Σ (常数) = n × 常数。
ts must be simplified to a factorised form whenever possible. Summation techniques are often tested together with proof by induction, where you prove that a given formula holds for all positive integers.
处理 sigma 表示的级数时，可以将求和式拆分为多个部分。例如 Σ (2r³ − 3r + 1) = 2 Σ r³ − 3 Σ r + Σ 1。结果应尽可能化为因式分解的形式。求和技巧常与归纳证明结合考查，要求证明给定的公式对所有正整数成立。
6. Numerical Methods | 数值方法
ton-Raphson method for an equation f(x) = 0 uses the iterative formula xₙ₊₁ = xₙ − f(xₙ)/f ' chosen near the root, and the process is repeated until the required accuracy is reached.
数值方法提供了一种在无法求得方程的代数解时求近似解的途径。对于方程 f(x) = 0，牛顿-拉夫森法使用迭代公式 xₙ₊₁ = xₙ − f(xₙ)/f '(xₙ)。在根附近选取合适的初始值 x₀，重复迭代直至达到所需精度。
ing the equation into the form x = g(x). The iteration xₙ₊₁ = g(xₙ) converges to a root if |g'(x)| ms and to discuss the failure of convergence when the gradient condition is not satisfied.
另一种方法是将方程重新排列为 x = g(x) 的形式进行迭代。若在根的邻域内满足 |g'(x)| < 1，则迭代 xₙ₊₁ = g(xₙ) 将收敛到该根。考试可能会要求用蛛网图或阶梯图展示迭代过程，并讨论当梯度条件不满足时收敛失败的情况。
7. Vectors in Three Dimensions | 三维向量
Vectors in 3‑ space are expressed in component form xi + yj + zk or as column vectors. The magnitude is |v| = √x² + y² + z². The scalar (dot) product a·b = |a||b| sine of the angle between two vectors and is used for checking perpendicularity (a·b = 0).
三维空间中的向量用分量形式 xi + yj + zk 或列向量表示。模为 |v| = √x² + y² + z²。标量积（点积）
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